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Full text of "Treatise On Analysis Vol-Ii"

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290 XIV INTEGRATION IN LOCALLY COMPACT GROUPS which is (14.10.7.1). If both/and g are in Jfc(G)» then so is/*#; since the formula (14.10.7.1) shows that the bilinear mapping (f,g)^f*g of &l(G, j8) x &l(G, j?) into «C(G) is continuous (5.5.1), and since jfc(G) is dense in &&G,P) and in ^c(G,j§) (13.11.6), the values of the mapping (j9 #)h->/* # belong to the closure of Jf C(G) in #C(G) (3.11.4): that is, they belong to <4(G) (13.20.5). Proposition (14.10.7) implies the following corollary: (14.10.8) Let A and B be two p-integrable sets in G. Then the function XH-+j3(A n XB) w continuous on G tf/w/ tewcfe to 0 at infinity (13.20.6). IfE"1 is also p-integrable, the function jcn-»/?(A n xB) is fl-integrable, and we have // moreover neither A nor B is ^negligible, the set AB * /z&s a nonempty interior. Finally, for every subset A wWcA is ^-measurable and not ^negligible, the set AA"1 is a neighborhood of the neutral element e ofG. We have <pA e &2(G, ]8) and «pB e ^f2(G, j8), hence we may apply (14.10.7) with /=<pA and # = <PB- Since ^B(5"Jx) = <pB(^"^) = ^B^). we have (^A * $B)(X) = f ^A n xflW ^(^) = j5(A n ^B). This proves the first assertion. If B"1 is also /Mntegrable, then both <pA and ^B-i belong to JSP^G, )8), and we can apply (14.10.6.2), which gives us the formula j /?(A n xE) dft(x) = ]8(A)jS(B"1). If the right-hand side of this formula is nonzero, it follows that /?(A n xB) is not identically zero, and hence there exists a nonempty open set U in which the continuous function jS(A n *B) is >0. This implies that U G AB""1. Finally, to prove the last assertion, we observe that there exists a compact set K c A which is not ^-negligible, so that we may assume that A is compact; but then j?(A) = (<pA * $A)(€) *s >^» an(^ ^ follows as above that there exists a neighborhood of e contained in AA"1. (14.10.9) Let /, g be two locally fi-integrable functions, (J. a measure on G. The two following conditions are equivalent: (a) fandg are convolvable and the function |/|* |^f| is p-integrable: (b) ju and f are convolvable and the function g(\ji\ * |/|) is fi-mtegrable. If these conditions are satisfied, we have (14.10.9.1) <0, 0*(/'/9> = </*&/*>• From (13.21.9) and (14.8.1), it follows that condition (a) means that thended.) Give an example in which the